Many students these days are confused by square roots. This is because there are three separate notions– the notion of square roots themselves, and the notion of the square root function, and the notion of “positive or negative square root”– and people tend to confuse all three of these different concepts.

A square root of x is a number whose square is x. Whenever x is positive, it has two square roots, one of which is positive, and one of which is negative. By abuse of English, we also speak of the square root of x, which always refers to the non-negative square root.

For example, 4 has two square roots, 2 and -2. This is because 2²=4 and (-2)²=4. But the square root of 4 is 2 (never -2).

The symbol √x, commonly pronounced “square root of x”, stands for the square root of x, i.e., the non-negative one. Thus, √4=2. √4 is never equal to -2, even though -2 is also a square root of 4.

As an example of why this matters, recall that the Pythagorean Theorem says that if a right triangle has a hypotenuse with length C, and two other sides length A and B, then C²=A²+B². This is often rephrased as C=√(A²+B²), which means, C is the positive square root of A²+B². C=-√(A²+B²) would also satisfy the equation C²=A²+B², but it would not make sense geometrically, since the hypotenuse can’t have negative length.

Another concept which makes things even more complicated is the plus-or-minus square root. There is actually no such thing as a plus-or-minus square root: it is merely language used to save words. For example, the equation x²=9 has two solutions, which are x=3 and x=-3. It is tedious to say: “since x²=9, x must be 3 or -3″. As a shortcut, we say: “since x²=9, x must be ±3.”

A very simple exercise goes like this. “Let f(x)=√x. Find f(25).” The answer is f(25)=√25=5. A surprisingly large number of students will say f(25)=±5. Remember, though, a function is only allowed to have one output for each input. Thus, f(25) can only have one output for 25.

FURTHER READING

How to Be Better at Algebra
Ambiguities in Mathematics
Self-Taught Calculus